Regarding the facet dependency on v3.0, there is an existing bug for this. See [1].

You do have to deploy your project to the server explicitly, to bring up the Axis2 servlet. Select Run -> Run As -> Run on Server. You can get the URL of the service after you click on the Services link.

If you want to test the web service, move the slider to the Test position in the web service wizard. Or, you could launch the WSE after the fact, and provide the URL of the service (WSDL) that you want to test. Go through this tutorial [2] to see if it addresses all your concerns.

I notice that under Project Facets it says Axis2 v 1.1 and I also had to
switch Dynamic Web Modules from v 3.0 to v 2.5. Might this be the source
of my problems?

Do I need to deploy the axis2 stuff on tomcat explicitly?

When building the WS (with axis2 dutifully selected); I get no error. But
there is no web-service explorer coming up. And, unlike for the case of
axis1 built WS, I don't seem to
find the magic wand to hook-up my new web-service with the web-service
explorer. which file would I select before I say "Web Services -> Test
with Web Services Explorer"?